BARI-TH/99-368 CERN-TH/99-403 UGVA-DPT 1999/11-1057 0 B Meson Transitions into Higher Mass Charmed 0 0 Resonances 2 n a J P. Colangeloa, F. De Faziob 1 and G. Nardullia,c,d 9 1 a Istituto Nazionale di Fisica Nucleare, Sezione di Bari, Italy 1 v b D´epartement de Physique Th´eorique, Univ. de Gen`eve, Switzerland 0 c Dipartimento di Fisica, Universit´a di Bari, Italy 0 2 d Theory Division, CERN, Gen`eve, Switzerland 1 0 0 0 / h p - Abstract p e We use QCD sum rules to estimate the universal form factors describing the h semileptonic B decays into excited charmed resonances, such as the 1− and : v 3− i 2− states D∗ and D∗ belonging to the sP = heavy quark doublet, and the X 1 2 ℓ 2 r 5− a 2− and 3− states D∗′ and D belonging to the sP = doublet. 2 3 ℓ 2 1“Fondazione Angelo Della Riccia” Fellow. Address after December 1st, 1999: Centre for Particle Physics, Durham University, United Kingdom 1 Introduction In the heavy quark (Q = c,b) infinite mass limit (m ) Quantum Chromodynamics Q → ∞ exhibits symmetries that are not present in the finite mass theory: heavy quark spin and flavour symmetries [1], as well as the velocity superselection rule [2]. These approximate symmetries allow to organize the spectrum of physical states comprising one light anti- quark andone heavy quark in multiplets of definite parityP and totalangular momentum s of the light degrees of freedom. ℓ The lowest lying multiplet consists inthemeson doublet withsP = 1−, corresponding ℓ 2 to the vector 1− and the pseudoscalar 0− state. The doublet can be described by a 4 4 × Dirac matrix (1+/v) H = [P∗γµ Pγ ] (1) 2 µ − 5 where v is the heavy meson velocity, P∗µ and P are annihilation operators of the 1− and a a 0− Qq¯ mesons (a = 1,2,3 for u,d and s); for charm, they are D∗ and D, respectively. 2 a The nearest mass multiplets are the sP = 1+ doublet, comprising the positive parity ℓ 2 1+ and 0+ states, and the sP = 3+ doublet which includes the positive parity 1+ and 2+ ℓ 2 states. In the charm sector three of such states have been identified: the state D (2460) 2 is the narrow 2+ meson with sP = 3+; moreover, there are two 1+ mesons with masses ℓ 2 mD10 = (2422.2 ± 1.8) MeV [3] and mD1∗0 = (2461+−4314 ± 10 ± 32) MeV [4]; they can be identified with members of the multiplets predicted by the Heavy Quark Effective Theory [5], including some mixing between them. Evidence for such states has also been collected in the beauty sector [6]. From the theoretical viewpoint these states have been the subject of intense scrutiny: the role of the 1+ doublet (0+, 1+) in some applications 2 of chiral perturbation theory has been considered in [5] and in [7]; their properties have been studied both by QCD sum rules [8, 9, 10] and quark models [11]. In this letter we investigate some properties of the next heavy meson multiplets, the sP = 3− doublet including two mesons with JP = 1− and 2−, and the sP = 5− doublet ℓ 2 ℓ 2 which comprises the states with JP = 2− and 3−. We estimate the universal form factors describing, in the infinite heavy quark mass limit, the semileptonic B decays into such multiplets, and consider the contribution of these processes to the inclusive semileptonic B decay width 3. 2The operators in (1) have dimension 32 since they contain a factor √mP in their definition. 3A review on the problems related to inclusive and exclusive semileptonic B decays can be found in ref.[12]. 1 WefollowtheQCDsumruleapproach[13], whichhasbeenappliedtosimilar problems in the past [8, 10, 14] 4. However, as discussed in the following, in the application of the methodtohigh-spinstatesseveraldifficultiesappearinidentifyingtherangeofparameters needed in the sum rule analyses, due to the peculiar features of the considered states and of their interpolating currents. In order to overcome such difficulties, we make use of information coming from other theoretical approaches, namely constituent quark models predicting the heavy meson spectrum. The final result, although affected by a sizeable theoretical uncertainty, nevertheless is useful for assessing the role of high-spin meson doublets in constituting part of the charm inclusive semileptonic B decay width. 2 Effective meson operators and quark currents The effective operators describing the sP = 3− and sP = 5− meson doublets are given ℓ 2 ℓ 2 respectively by [7]: 1+/v 3 γ Hµ = D∗µνγ γ D∗ν(gµ ν(γµ +vµ) (2) 2  2 5 ν −s2 1 ν − 3    1+/v 5 γ γ Hµν = Dµνσγ γ D∗′αβ gµgν αgν(γµ vµ) βgµ(γν vν) , (3) 2  3 σ −s3 5 2 α β − 5 β − − 5 α −  (cid:18) (cid:19)   where D∗ represent annihilation operators of the mesons with appropriate quantum num- i bers. In order to implement the QCD sum rule programme, we need quark currents with non-vanishing projection on these states. They have been investigated in ref.[10] and are given by the following expressions: 3 γα sP = (3)−; JP = 1− : Jα = h¯ Dα t D/ q (4) ℓ 2 − vs4 t − 3 t (cid:20) (cid:21) 1 sP = (3)−; JP = 2− : Jαβ = Tαβ,µνh¯ γ γ D q (5) ℓ 2 v "√2 5 tµ tν# 5 2 sP = (5)−; JP = 2− : J˜αβ = Tαβ,µνh¯ γ D D D γ D/ q(6) ℓ 2 −s6 v 5 tµ tν − 5 tµ tν t (cid:20) (cid:21) i sP = (5)−; JP = 3− : Jαβλ = Tαβλ,µνσh¯ γ D D q , (7) ℓ 2 v"√2 tµ tν tσ# where Dµ is the covariant derivative: Dµ = ∂µ igAµ , and Gµ represents the t − transverse component of the four-vector Gµ with respect to the heavy quark velocity v: 4For a review see [15]. 2 Gµ = Gµ (G v)vµ . The tensors Tαβ,µν and Tαβλ,µνσ are needed to symmetrize indices t − · and are given by 1 1 Tαβ,µν = gαµgβν +gανgβµ gαβgµν (8) 2 − 3 t t 1 (cid:16) (cid:17) Tαβλ,µνσ = gαµgβνgλσ +gανgβµgλσ +gασgβνgλµ 3 1 (cid:16) (cid:17) gαβgµνgλσ +gαλgµνgβσ +gβλgµνgασ , (9) − 3 t t t t t t t t t (cid:16) (cid:17) with gαβ = gαβ vαvβ . t − As discussed in [10], in the m limit the currents in eqs.(4)-(7) have non- Q → ∞ vanishing projection only to the corresponding states of the HQET, without mixing with states of the same quantum number but different s content. Therefore, we can define a ℓ set of one-particle-current couplings as follows: sPℓ = (23)−; JP = 1− : < D1∗(v, ǫ) |Jα| 0 > = f1 √mD1∗ ǫ∗α (10) sPℓ = (32)−; JP = 2− : < D2∗(v, ǫ) |Jαβ| 0 > = f2 √mD2∗ ǫ∗αβ (11) sPℓ = (52)−; JP = 2− : < D2∗′(v, ǫ) |J˜αβ| 0 > = f˜2 √mD2∗′ ǫ∗αβ (12) sPℓ = (52)−; JP = 3− : < D3∗(v, ǫ) |J˜αβλ| 0 > = f3 √mD3∗ ǫ∗αβλ , (13) where ǫ are the meson polarization tensors. The couplings f are low-energy parameters, i determined by the dynamics of the light degrees of freedom. Since the two pairs (f , f ) 1 2 ˜ ˜ and (f , f ) are related by the spin symmetry, in the sequel we only consider f and f . 2 3 1 2 3 Two-point function sum rules ˜ To evaluate the parameters f and f let us consider the two-point correlators 1 2 i d4x e−iωv·x < 0 T(J†α(x)Jβ(0) 0 > = Π (ω)gαβ (14) 1 t | | Z 2 i d4x e−iωv·x < 0 T(J˜†αβ(x)J˜µν(0) 0 > = Π (ω) gαµgβν + gανgβµ gαβgµν | | 2 t t t t − 3 t t Z (cid:18) (cid:19) (15) given in terms of Π and Π , scalar functions of the variable ω. 1 2 As extensively discussed in the literature, the QCD sum rule method amounts to eval- uate the correlators in two equivalent ways. On one side the Operator Product Expansion 3 (OPE) is applied for negative values of ω; the expansion produces an asymptotic series, whose leading term is the perturbative contribution (computed in HQET), followed by subleading terms parameterized bynonperturbative quantities, suchasthequarkconden- sate: < q¯q >, the gluon condensate: < α G Gµν >, the mixed quark-gluon condensate, s µν etc. On the other side, one evaluates the correlators by writing down dispersion relations (DR) for the scalar functions Π (ω) and Π (ω); they get contributions by the hadronic 1 2 states, in particular by the low-lying resonances with appropriate quantum numbers. To get rid of radial excitations and multiparticle states, one performs a Borel transform on both sides of the sum rule, which enhances the low mass contribution of the spectrum; moreover, assuming quark-hadron duality, one identifies, from some effective continuum threshold ω , the hadronic side of the sum rules with the perturbative result obtained by c the OPE. In the final sum rule, only the contributions from the physical to the continuum threshold appear: the low mass resonance on one side, the OPE truncated at ω on the c other. Applying the method to the correlators (14) and (15) we get two borelized sum rules ˜ for the parameters f and f : 1 2 2 ω1c < q¯σGq > f2e−∆1/E = dσ σ4e−σ/E + (16) 1 π2 16 Z0 f˜2e−∆2/E′ = 16 ω2c dσ σ6e−σ/E′ . (17) 2 5π2 Z0 Here the parameters ∆1 and ∆2 are defined by the formulae: ∆1 = mD1∗ −mc and ∆2 = mD2∗′−mc,mc being thecharmquarkmass; therefore, theparameters∆1 and∆2 represent the binding energy of the states D∗ and D∗′, which is finite in the infinite heavy quark 1 2 mass limit. On the other hand, ω and ω represent the effective thresholds separating 1c 2c the low-lying resonances from the continuum; E and E′ are parameters introduced by the Borel procedure. Relations for the mass parameters ∆ and ∆ can be obtained by 1 2 taking derivatives of the sum rules (16) and (17): 1 ω1c dσ σ5e−σ/E π2 ∆ = Z0 (18) 1 1 ω1c < q¯σGq > dσ σ4e−σ/E + π2 32 Z0 ω2cdσ σ7e−σ/E′ ∆ = Z0 . (19) 2 ω2cdσ σ6e−σ/E′ Z0 There is an important point deserving a discussion, and it concerns the high dimen- sionality of the interpolating currents Jα and J˜αβ, which has two consequences on the 4 structureofthesumrules(16)-(17)and(18)-(19). First, thespectralfunctionsineqs.(16)- (17) and (18)-(19) have large powers, and therefore the perturbative contributions in the sum rules are very sensitive to the continuum thresholds ω and ω . The second ef- 1c 2c fect consists in the absence of the contributions from low-dimensional condensates, which implies (neglecting high-dimensionalcondensates) completedualitybetween theperturba- tive and the hadronic contributions to the sum rules. Such two effects cannot be avoided in our analysis, and are typical of the sum rule approach to high spin states [16]. In our case they have the main consequence of not allowing to determine simultaneously the couplings f andthe mass parameters ∆ , due tothe critical dependence on thecontinuum i i thresholds. Therefore, we adopt the strategy of getting the values of the mass parameters from other determinations, and then to fix the thresholds from eqs.(18)-(19) and comput- ing f from (16)-(17). Admittedly, this is a hybrid procedure, which nevertheless allows i us to estimate both the current-particle matrix elements and the universal semileptonic form factors, as discussed in the next Section. WhileexperimentalinformationonthesP = 3− andsP = 5− doubletsarenotavailable ℓ 2 ℓ 2 so far, there are studies concerning such states based on constituent quark models [17]. They suggest that the mass of the 3− (cu¯) state D is m = 2.83 GeV or m = 2.76 3 D3 D3 GeV, whereas the mass of the corresponding (bu¯) state is m = 6.11 GeV. Assuming a B3 spin splitting of 40 MeV in the charm sector, as suggested by the same models, we can ≃ give to the mass of the 5− state the value of 2.78 GeV, e.g. nearly 0.8 GeV above the 0− 2 doublet (the same value comes from the analysis of the beauty meson spectrum). This implies for the parameter ∆ a value in the range ∆ [1.2 1.4] GeV, considering the 2 2 ≃ − determination of the analogous binding energy of B and D mesons [15]. As for ∆ , we fix 1 it to ∆ [1.2 1.4] GeV, according to similar considerations. 1 ≃ − Let us consider ∆ and ∆ related to the thresholds ω and to the Borel parameters 1 2 i E by eqs.(18)-(19). There is a range of Borel parameters and thresholds where the i chosen binding energies can be obtained. In particular, while the dependence of ∆ on i the Borel parameters is quite mild, so that the range E = [1 1.5] GeV can be chosen, i − the dependence on the thresholds, as expected, is critical: one has to choose ω in a quite i narrow range [1.6 1.8] GeV to obtain ∆ . However, this choice is not unappropriate, i − since it suggests that the continuum threshold is above the mass of the corresponding resonance by nearly the mass of one pion. After having fixed ∆ and the ranges of E and of ω , from eqs.(16)-(17) we can obtain i i i 5 ˜ 7 the values of the couplings fi: f1 = [0.6 0.8] GeV2 and f2 = [1.2 1.6] GeV2. Notice − − 5 that, at odds, e.g., with the leptonic constants related to the matrix elements of the quark axial currents on the 0− state, the couplings f do not have an immediate physical i meaning, as they represent the projections of the interpolating currents on the orbitally excited meson states. Nevertheless, they play an important role in the determination of the form factors, as we discuss in the next Section. 4 Universal form factors from three-point sum rules There are two universal form factors describing the semileptonic B decays into the excited negative parity charmed resonances with s = 3− and s = 5−. The first one, τ , governs ℓ 2 ℓ 2 1 the decays B D∗ℓν (20) → 1 ℓ B D∗ℓν (21) → 2 ℓ in the heavy quark limit. The second one, τ , describes in the same limit the decays 2 B D∗′ℓν (22) → 2 ℓ B D ℓν . (23) 3 ℓ → It is straightforward to write down the semileptonic matrix elements for the transitions (20)-(23), by applying, e.g., the trace formalism [15]. One obtains: 3 1 y < D1∗(v′,ǫ)|(V −A)µ|B(v) > = √mBmD1∗ τ1(y) s2 (ǫ∗v) vµ −v′µ + −3 v′µ h (cid:18) (cid:19) 1 y2 1 y − ǫ∗µ +i − ǫµλρσǫ∗v′v , (24) − 3 3 λ ρ σ i < D2∗(v′,ǫ)|(V −A)µ|B(v) > = √mBmD2∗ τ1(y) ǫ∗λν vλ gµν(y −1)−vνv′µ + i ǫαβνµv′ v h (25) α β i for the decays (20) and (21), while for the decays (22) and (23) the relevant matrix elements can be written as: 5 2(1 y2) < D2∗′(v′,ǫ)|(V −A)µ|B(v) > = s3 √mBmD2∗′ τ2(y) ǫ∗αβ vα −5 gµβ −vβvµ h 2y 3 2(1+y) + − vβv′µ + i ǫµλβρv v′ , (26) 5 5 λ ρ i < D (v′,ǫ) (V A)µ B(v) > = √m m τ (y) ǫ∗ vαvβ gµλ(1+y) vλv′µ 3 | − | B D3 2 αβλ − + iǫµλρτv v′ . h (27) ρ τ i 6 In these equations the weak current is (V A)µ = c¯γµ(1 γ )b, y = v v′ and τ (y), τ (y) 5 1 2 − − · are the universal form factors. At the zero-recoil point v = v′ the matrix elements in (24)-(27) vanish, as expected by the heavy quark symmetry. As a matter of fact, for B decays into spin 2 and spin 3 states, at least one index of the final meson polarization tensor is contracted by the B four-velocity v, and therefore the product vanishes for v = v′. The spin symmetry requirement being verified in the matrix elements, the Isgur-Wise form factors τ and τ 1 2 are not required to vanish at v v′ = 1. · Onecan attempt anestimate ofthe formfactorsτ by three-point function sum rules, 1,2 considering the correlators (relevant for the matrix elements (24) and (26)): i2 d4x d4z e−i(ωv·x−ω′v′·z) < 0 T(J†α(z)(V A)µ(0)J (x) 0 > = 5 | − | Z = iǫµαβλv v′Ω (ω, ω′)+Ξ (ω, ω′)wαvµ + ... (28) β λ 1 1 i2 d4x d4z e−i(ωv·x−ω′v′·z) < 0 T(J˜†αβ(z)(V A)µ(0)J (x) 0 > = 5 | − | Z = iǫµστρv v′(wαgβ + wβgα)Ω (ω, ω′) + wαwβvµΞ (ω, ω′) + ...(29) τ ρ σ σ 2 2 where wα = vα yv′α, J = q¯iγ b; the dots represent other Lorentz structures which are 5 5 − not relevant for the subsequent analysis, since we only consider Ω and Ω . 1 2 Since the scalar functions Ω depend on two variables, one has to perform double DRs j and double Borel transforms, which introduces, for each sum rule, two Borel parameters E and E′. The resulting equations read: τ (y) = 9 e∆/E+∆1/E′ ωc ω1c dσdσ′ e−σ/E−σ/E′h (σ, σ′)θ(σ, σ′) (30) 1 2√2π2f1Fˆ Z0 Z0 1 τ (y) = 3 e∆/E+∆2/E′ ωc ω2cdσdσ′ e−σ/E−σ/E′h (σ, σ′)θ(σ, σ′) (31) 2 √2π2f˜2Fˆ Z0 Z0 2 where 1 σ2 +σ′2 2yσσ′ σ′(σ +σ′) h (σ, σ′) = − + 1 (y2 1)3/2 " 2(y 1) 3 # − − 1 h (σ, σ′) = 5σ3 3σ′σ2(4y 1)+(2y2 2y +1)(σ′3 +3σσ′2) , 2 (y +1)(y2 1)5/2 − − − − h i (32) and χ(σ, σ′) = Θ(σ2 +σ′2 2yσσ′) , (33) − 7 with Θ(x) the step function. In eqs.(30) and (31) the parameter ∆ represents the mass difference between the low − lying multiplet s = 1 and the heavy quark. The integration region can be expressed ℓ 2 in terms of the varia(cid:16)ble(cid:17)s σ +σ′ σ = + 2 σ σ′ σ = − − 2 and one can choose the triangular region defined by the bounds: 0 σ ω(y) (34) + ≤ ≤ y 1 y 1 − σ σ + − σ . (35) + − + −sy +1 ≤ ≤ sy +1 As to the upper limit in the integration interval for ω we adopt + ω +ω 1c c ω(y) = (36) 2 1+ y−1 y+1 (cid:16) q (cid:17) for the two cases studied in this letter (we use, according to the two-point sum rule analysis ω = ω ). c1 c2 We use the value Fˆ = 0.21 GeV3/2, which is obtained by QCD sum rules [8, 15] with α = 0 (the same order which we consider in the present analysis). Moreover, we use s ∆ = 0.5 GeV, with the threshold in the B channel ω = 0.7 GeV. As for the charm c channel, we use ω = ω = 1.6 1.8 GeV. 1c 2c − We can now numerically determine the form factors τ , using the above equations. i The result for the universal function τ (y), obtained within the uncertainties discussed 1 above, is that this function, in the whole kinematical region relevant for the decays (20)- (21), is less than 10−4, which implies that, in the infinite heavy quark mass limit, the semileptonic B transitions into the s = 3− doublet have a very small decay width. The ℓ 2 situation is different for the universal function τ (y), which is depicted in fig.4 where the 2 shaded region corresponds to the results obtained by varying the parameters ∆, ∆ , ω 2 c and ω in the ranges quoted above. The form factor τ , at the zero recoil point y = 1, 2c 2 is in the range τ (1) = 0.10 0.20, with a mild y dependence that can be neglected, 2 − − within the accuracy of the sum-rule method. Although it is difficult to reliably assess the theoretical accuracy of this result, it is interesting to observe that a form factor in the range quoted above implies that the semileptonic channel is experimentally accessible. 8 0000....3333 0000....2222 0000....1111 0000 1111....00005555 1111....1111 1111....11115555 1111....2222 Figure 1: Universal form factor τ (y) 2 5 Semileptonic decay rates Using the parameterization of the B matrix elements in eqs.(26) and (27) we can work out the expressions of the widths of the decay modes (22) and (23), which are respectively given by: dΓ(B D∗′ℓν ) = G2FVc2bm2Bm3D2∗′(τ (y))2(y 1)25(y+1)27[(1+r2)(7y 3) 2r(4y2 3y+3)] dy → 2 ℓ 720π3 2 − − − − (37) dΓ(B D ℓν ) = G2FVc2bm2Bm3D3(τ (y))2(y 1)52(y+1)72[(1+r2)(11+3y) 2r(11y+3)] dy → 3 ℓ 720π3 2 − − (38) with r = mmDBi. Using mD3 = 2.78 GeV, mD2∗′ = 2.74 GeV and τ2(y) = 0.15, we get Γ(B D∗′ℓν ) Γ(B D ℓν ) 4 10−18 GeV (39) → 2 ℓ ≃ → 3 ℓ ≃ × and B(B D∗′ℓν ) B(B D ℓν ) 1 10−5 . (40) → 2 ℓ ≃ → 3 ℓ ≃ × 9